O monitoramento da média com mínimos quadrados suporta a descrição de dados vetoriais

Maboudou-Tchao, Edgard M.

Edgard M. Maboudou-Tchao

Department of Statistics & Data Science, University of Central Florida, Orlando, FL, USA. E-mail: edgard.maboudou@ucf.edu.

http://orcid.org/0000-0001-6783-282X

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Figure 1
Boundary Shapes for LS-SVDD, (top, left) C = 0.5 and σ = 10.5, (top right) C = 10 and σ = 1.5, (bottom, left) C = 50 and σ = 1.5, (bottom, right) C = 500 and σ = 1.5.

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Figure 2
LS-SVDD and SVDD charts for the Breast cancer dataset example. (left) LS-SVDD chart, (right) SVDD chart.

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Table 1
Performance comparison of the charts for Multivariate Normal – ARL (standard errors are shown inside the parentheses), (left) p = 5, (right) p = 10.

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Table 2
Performance comparison of the charts for Multivariate t with 3 degrees of freedom – ARL (standard errors are shown inside the parentheses), (left) p = 5, (right) p = 10.

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Table 3
Performance comparison of the charts for Multivariate Lognormal – ARL (standard errors are shown inside the parentheses), (left) p = 5, (right) p = 10.

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Table 4
Impact of the Sample size – ARL (standard errors are shown inside the parentheses), p = 5.

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Table 5
Impact of the Sample size – ARL (standard errors are shown inside the parentheses), p = 10.

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Charts, p = 5				Charts, p = 10
δ	LS-SVDD	SVDD	T2	δ	LS-SVDD	SVDD	T2
0	200.3 (1.651)	200.1 (1.642)	202.8 (1.61)	0	201.5 (1.65)	199.6 (1.62)	201.8 (1.651)
0.1	194 (1.556)	192.1 (1.584)	201.8 (1.671)	0.1	194.8 (1.608)	195 (1.591)	199.7 (1.62)
0.2	180.2 (1.475)	180.5 (1.465)	200.5 (1.646)	0.2	190.3 (1.533)	191.7 (1.594)	195.2 (1.567)
0.3	166.5 (1.358)	166.2 (1.337)	198.9 (1.634)	0.3	176.9 (1.439)	180.2 (1.45)	187.8 (1.531)
0.4	149.9 (1.216)	151.6 (1.232)	190.4 (1.54)	0.4	162 (1.309)	168.9 (1.362)	184.3 (1.507)
0.5	131.7 (1.069)	133.2 (1.07)	173.1 (1.418)	0.5	153 (1.231)	156.3 (1.262)	174.8 (1.436)
0.6	114.9 (0.9353)	117.6 (0.9608)	156.5 (1.27)	0.6	140.3 (1.142)	143.6 (1.173)	161.4 (1.305)
0.7	99.21 (0.8095)	104.8 (0.8424)	140.7 (1.128)	0.7	124.7 (1.015)	130.6 (1.063)	149.1 (1.207)
0.8	84.3 (0.6843)	89.38 (0.7232)	123.5 (1.01)	0.8	111.5 (0.8993)	116.5 (0.9506)	139.3 (1.14)
0.9	71.68 (0.5658)	75.57 (0.616)	106.2 (0.8754)	0.9	97.06 (0.7921)	101.5 (0.8192)	125.42 (1.0 16)
1	60.53 (0.4889)	63.36 (0.5102)	93.19 (0.7584)	1	88.12 (0.7205)	88.7 (0.7163)	112.1 (0.9194)

Charts, p = 5				Charts, p = 10
δ	LS-SVDD	SVDD	T2	δ	LS-SVDD	SVDD	T2
0	199.1 (1.612)	200.1(1.675)	198.5 (1.611)	0	199.3 (1.626)	201.9 (1.658)	199 (1.607)
1	200.1 (1.623)	201.1 (1.697)	195.5 (1.589)	1	199.1 (1.622)	201.1 (1.635)	201.9 (1.65)
2	197.8 (1.614)	200.8 (1.67)	193.3 (1.556)	2	196.8 (1.596)	200.6 (1.62)	195.4 (1.579)
3	189.1 (1.544)	200.6 (1.653)	184.9 (1.48)	3	194 (1.575)	194.9 (1.575)	189.9 (1.558)
4	183.5 (1.512)	188.3 (1.531)	176.1 (1.443)	4	186 (1.496)	191.2 (1.573)	180 (1.468)
5	169.5 (1.377)	181.1 (1.464)	162.7 (1.306)	5	179.7 (1.445)	182.1 (1.484)	175.2 (1.415)
6	159.2 (1.291)	169.3 (1.361)	150.7 (1.226)	6	170.2 (1.384)	178.2 (1.434)	162.3 (1.313)
7	140.2 (1.139)	151.8 (1.238)	133.7 (1.101)	7	160 (1.318)	168.6 (1.373)	148.6 (1.209)
8	122.2 (0.9952)	134.3 (1.087)	115.2 (0.933)	8	151.9 (1.221)	157.1 (1.277)	134.9 (1.089)
9	100.9 (0.8227)	111.8 (0.9191)	98.63 (0.8058)	9	136.8 (1.102)	146.9 (1.181)	118.1 (0.9529)
10	80.80 (0.6504)	92.66 (0.7525)	80.08 (0.6485)	10	121.7 (0.9961)	131.8 (1.053)	102.4 (0.831)

Charts, p = 5				Charts, p = 10
δ	LS-SVDD	SVDD	T2	δ	LS-SVDD	SVDD	T2
0	199.1 (1.613)	201(1.646)	202.8 (1.641)	0	199.9 (1.609)	201.4 (1.654)	199 (1.607)
0.1	178.4 (1.469)	175.5 (1.414)	176.2 (1.439)	0.1	187.7 (1.535)	187.9 (1.523)	188.1 (1.547)
0.2	156.6 (1.281)	153.5 (1.248)	151.9 (1.236)	0.2	170.8 (1.393)	169.3 (1.39)	171.3 (1.396)
0.3	136.7 (1.098)	129 (1.042)	133.2 (1.088)	0.3	155.9 (1.271)	154.3 (1.255)	154.6 (1.255)
0.4	115.8 (0.9386)	110.1 (0.8888)	111.5 (0.914)	0.4	140.1 (1.142)	135.5 (1.109)	136 (1.113)
0.5	97.72 (0.7888)	90.84 (0.745)	94.56 (762)	0.5	125.7 (1.023)	119.9 (0.9781)	120.7 (0.9829)
0.6	83.87 (0.6676)	74.84 (0.6061)	78.61 (0.6332)	0.6	112.2 (0.9167)	106 (0.8701)	106.7 (0.8655)
0.7	68.02 (0.5489)	62.83 (0.5064)	66.59 (0.5359)	0.7	95.55 (0.7829)	90.06 (0.7174)	93.75 (0.756)
0.8	56.21 (0.4531)	50.71 (0.4079)	55.43 (0.4417)	0.8	83.29 (0.6778)	77.24 (0.6326)	80.97 (0.6652)
0.9	47.06 (0.38)	41.86 (0.3417)	45.33 (0.3698)	0.9	72.73 (0.5759)	65.71 (0.5379)	69.99 (0.5665)
1	39.51 (0.3247)	34.33 (0.2742)	37.97 (0.305)	1	61.22 (0.4953)	55.46 (0.4569)	59.74 (0.485)

Sample size, p = 5
δ	N = 30	N = 50	N = 75	N = 90	N = 100
0	200.5 (1.629)	201.5 (1.656)	201.3 (1.63)	199.3 (1.619)	200.3 (1.651)
0.1	196.7 (1.594)	188.8 (1.535)	191.6 (1.549)	189.5 (1.549)	194 (1.556)
0.2	187.4 (1.524)	172.7 (1.416)	179.2 (1.453)	179.4 (1.458)	180.2 (1.475)
0.3	175.6 (1.434)	157.6 (1.271)	162.7 (1.34)	165.9 (1.353)	166.5 (1.358)
0.4	158.1 (1.297)	140.6 (1.14)	147.5 (1.192)	145.1 (1.189)	149.9 (1.216)
0.5	143.1 (1.16)	120.7 (0.9761)	127.9 (1.037)	131 (1.058)	131.7 (1.069)
0.6	126.1 (1.032)	103.3 (0.8459)	113 (0.9194)	114.4 (0.9322)	114.9 (0.9353)
0.7	109.8 (0.9006)	89.42 (0.7207)	95.17 (0.7678)	98.41 (0.8046)	99.21 (0.8095)
0.8	94.34 (0.7618)	74.46 (0.6041)	82.86 (0.6567)	83.93 (0.6867)	84.3 (0.6843)
0.9	80.76 (0.6551)	62.63 (0.5052)	68.3 (0.5513)	70.43 (0.5595)	71.68 (0.5658)
1	67.77 (0.5489)	52.99 (0.4264)	58.97 (0.478)	60.34 (0.4843)	60.53 (0.4889)

Charts, p = 10
δ	N = 30	N = 50	N = 75	N = 90	N = 100
0	198.6 (1.663)	202.7 (1.654)	202.5 (1.643)	199 (1.637)	199.6 (1.62)
0.1	211 (1.709)	204.3 (1.654)	199.6 (1.632)	201.6 (1.642)	195 (1.591)
0.2	216.5 (1.776)	204 (1.678)	192.2 (1.566)	192.5 (1.58)	191.7 (1.594)
0.3	221.4 (1.796)	202.2 (1.65)	183.3 (1.499)	185.3 (1.507)	180.2 (1.45)
0.4	218 (1.784)	191.8 (1.558)	169.5 (1.389)	174.9 (1.412)	168.9 (1.362)
0.5	215.7 (1.731)	182.7 (1.491)	157.8 (1.295)	160.9 (1.322)	156.3 (1.262)
0.6	211.3 (1.735)	171.4 (1.381)	142.8 (1.153)	144.4 (1.169)	143.6 (1.173)
0.7	199.2 (1.653)	155.9 (1.205)	128.1 (1.045)	130.1 (1.065)	130.6 (1.063)
0.8	184.2 (1.512)	139.8 (1.148)	116.1 (0.9488)	118.2 (0.9557)	116.5 (0.9506)
0.9	170.1 (1.38)	128.6 (1.034)	100.4 (0.8142)	105.3 (0.8544)	101.5 (0.8192)
1	158.1 (1.29)	112.7 (0.9111)	86.81 (0.7095)	92.3 (0.7377)	88.7 (0.7163)

\begin{array}{l} \begin{matrix} minimize \\ r, a, ξ \end{matrix} r^{2} + C \sum_{i = 1}^{N} ξ_{i}, \\ subject to φ (x_{i}) - a^{2} = {(φ (x_{i}) - a)}^{'} (φ (x_{i}) - a) \leq r^{2} + ξ_{i}, i = 1, 2, \dots, N \\ ξ_{i} \geq 0, i = 1, 2, \dots, N \end{array}

\begin{matrix} L (r, a, β_{i}, γ_{i}, ξ_{i}) = r^{2} + C \sum_{i = 1}^{N} ξ_{i} - \sum_{i = 1}^{N} β_{i} (r^{2} + ξ_{i} - (φ {(x_{i})}^{'} φ (x_{i}) - 2 φ {(x_{i})}^{'} a + a^{'} a)) - \sum_{i = 1}^{N} γ_{i} ξ_{i}, \\ = r^{2} (1 - \sum_{i = 1}^{N} β_{i}) + \sum_{i = 1}^{N} ξ_{i} (C - β_{i} - γ_{i}) + \sum_{i = 1}^{N} β_{i} (φ {(x_{i})}^{'} φ (x_{i}) - 2 φ {(x_{i})}^{'} a + a^{'} a) \end{matrix}

\begin{array}{l} \begin{matrix} minimize \\ R, c, ξ \end{matrix} R^{2} + C \sum_{j = 1}^{N} ξ_{j}^{2}, \\ subject to φ (x_{j}) - c^{2} = {(φ (x_{j}) - c)}^{'} (φ (x_{j}) - c) = R^{2} + ξ_{j}, j = 1, 2, \dots, N \\ ξ_{j} \geq 0, j = 1, 2, \dots, N \end{array}

\begin{array}{l} \max_{α} \sum_{j = 1}^{N} α_{j} φ {(x_{j})}^{'} φ (x_{j}) - \sum_{i, j = 1}^{N} α_{i} α_{j} (φ {(x_{i})}^{'} φ (x_{j}) + \frac{1}{2 C} δ_{i j}) \\ subject to \sum_{j = 1}^{N} α_{j} = 1 \end{array}

L (α) = \sum_{j = 1}^{N} α_{j} φ {(x_{j})}^{'} φ (x_{j}) - \sum_{i, j = 1}^{N} α_{i} α_{j} (φ {(x_{i})}^{'} φ (x_{j}) + \frac{1}{2 C} δ_{i j}) + γ (\sum_{j = 1}^{N} α_{j} - 1)

R^{2} = \frac{1}{N} \sum_{s = 1}^{N} (K (x_{s}, x_{s}) - 2 \sum_{j = 1}^{N} α_{j} K (x_{s}, x_{j}) + \sum_{j, l}^{N} α_{j} α_{l} K (x_{j}, x_{l}))

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O monitoramento da média com mínimos quadrados suporta a descrição de dados vetoriais

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